I FAIL at FOIL

[tzikeh]

Once upon a time I could figure this out in my sleep. Unfortunately, this is not once upon a time. Can any of you guys explain to me how to count up the following and get the right answer? (And no, this isn't a problem on a final exam or anything *g*)

You are a professor. You assign grades not by letter, but by x points per assignment (out of 100). Each assignment is worth y% of the final grade. Some are more (20% or 25%), some are less (5% or 10%).

If there were ten assignments worth 10% each, well, base ten is my friend and I wouldn't be asking this question. But there are seven assignments, worth 10%, 15%, 10%, 15%, 20%, 25%, and 5%.

You are totaling up each student's final grade for submission. What is a possible algebraic formula used to find their numeric grade, if there is one? If not, what kind of arithmetic do you use?

If you have the answer (or answers), can you explain to me how you arrived at it?

Comments

[taffimai.livejournal.com]

You multiply the grade on each assignment by its assigned weight and add up the results to get the students' final grade.

[tzikeh.livejournal.com]

So, let's say on the 15% one, the student has 86, you multiply 86 by 15%, and you do this for each assignment, and add up the sums?

If this is right, can you explain *how* you knew that answer... as if I were a student in a math class in which this particular type of math were being taught?

[taffimai.livejournal.com]

Yeah, exactly.

It just goes back to percentages. What you're doing is assigning each grade a weight, so you multiply it by its weight. You know this instinctively. If there were two assignments which were worth 50% each you'd take half (i.e. 50%) of each assignment's grade and add them together. This is just a more complex case.

[fiamaya.livejournal.com]

Hi -- I tripped over this looking at friendsfriends, but I teach this stuff, so i figured I'd chime in.

That is, absolutely, the right way to calculate it. Here's how I'd describe it. Say there are 100 points possible in the class. The 15% assignment is worth a total of 15 of those points. The student got 89% on that assignment, so she gets 89% of the possible 15 points, or .89*15=13.35 points.

FOr each assignment, you can do the same thing: it's worth 10 (15, 20, 25) points, and the students score is the percentage of the possible points on that assignment that she got. Add those all up, to get her score out of 100. If she'd gotten a perfect score on every assignment, she'd get the full 100. If you divide by that perfect 100, you get her weighted score as a percentage.

(Here's a bonus extra credit. Say three assignments -- the 20%, 25%, and one of the 15% ones -- are all exams, and the rest are homework and projects. Then her average score on the exams would be the total number of points she got on the exams, calculated like I said above, divided by the total possible number of points on the exams, or 20+25+15=60. That would give you her average percentage score on the exams, taking the weighting into account; someone who got a perfect score on every exam would have a 1, or 100%, as her average exam score.)

Does that help? Feel free to ask more, or ignore the stranger lecturing you.

[tzikeh.livejournal.com]

or ignore the stranger lecturing you.

Not at all - I asked for a lecture on how to solve it! :D Thank you very much for the unpacked explanation.

[fiamaya.livejournal.com]

Oh good. I'm almost all lurker on LJ, and when I do comment - particularly to explain something -- I get totally neurotic about being perceived as the obnoxious stranger barging in on a room full of friends.

[nomadicwriter]

So, let's say on the 15% one, the student has 86, you multiply 86 by 15%, and you do this for each assignment, and add up the sums?

Yeah, that's essentially right. It's maybe a little more intuitive to understand if you flip that equation around. Think of it as 86% of 15. If the student had scored 100/100, they would have earned 15 points for the assignment. Because they only scored 86/100, they only earned 86% of that possible 15 points; i.e. (15 * 0.86) or 12.9 points.

[isiscolo.livejournal.com]

Just multiply each assignment's points by the percentage, expressed as a decimal, i.e., 10% = .10, 15% = .15, and add them up. That's what percentages ARE.

[tzikeh.livejournal.com]

Hm.

Let's say, for shits and giggles, that you had to do it another way. Something slow and laborious. Could you, for example, take each assignment's point value, put it down for x% (e.g. an assignment worth 15% got 89 points, so you put down 89 (10%) plus 44.5 (5%) for 15% of the final sum, do this for each assignment, add it all up, and then divide by seven?

Or am I on some crack planet full of wtf math with that one?

[reginagiraffe]

By 89(10%) do you mean "89 times 10%"?

And why are you dividing by 7?

[isiscolo.livejournal.com]

I vote WTF math, because I totally can't even figure out what you are getting at here. Why is 89 10%? 89 is 10% of 890. So - what?

What information are you trying to get at, here? A bunch of people have answered you, so obviously it's not just the formula. Maybe if you said specifically what you are after, we'd be able to help you.

[tzikeh.livejournal.com]

Why is 89 10%?

89 points on an assignment (out of a possible 100) that's worth 10% of the final grade. The folks up above solved it for me. :D I thought I was clear -- sorry if sucked at explaining!

[tzikeh.livejournal.com]

The "slow and laborious" version was me trying to solve it *without* multiplying by percentages - like, if you didn't know about percentages, how else could you go about solving it?

[lamardeuse.livejournal.com]

If you're trying to teach it to someone who doesn't understand percentages, for example, an elementary school student, you could say that some marks are worth more than others. You then could determine that worth by looking at the LCD of the marks. In your example above, the LCD is 5%, so that means there are 20 parts or "chunks" to arrive at the total mark. Let's take your 89 example, worth 15%:

Start by dividing the mark by 20: that gives you the basic "chunk" of 5%:

89/20 = 4.45

Then, you see that this mark is worth 3 "chunks" (3 x 5 = 15), so you multiply it by 3:

4.45 x 3 = 13.35

You repeat this process with the 6 other marks, then add them up to get the total mark.

ETA: If you want to explain how you got the base 5, I suppose the student would have to understand LCD, or you could just say the mark weights are all part of the 5 x table if you wanted to simplify it. Also, 5 x 20 = 100, which is how you get the number of "chunks" in the total.

[fiamaya.livejournal.com]

You're trying to divide by seven because there are seven assignments, right? You can only do that to get the average if each assignment is worth the same amount. You calculate an average by multiplying each grade by the weight of that grade -- ie, the number of points that assignment is worth -- and dividing by the sum of the weights, or the total number of points. If each assignment is worth the same thing, that gives you the same result as adding up all of the scores and dividing by however many scores there are.

Say there are four assignments, on which a student scored 60%, 70%, 80%, and 90% respectively. If they're all worth the same amount, or 25% of the total, you could do it the hard way:
(.60*25+.7*25+.8*25+.9*25)/(25+25+25+25), but that's the same as
((.6+.7+.8+.9)*25)/(4*25), or the more common:
(.6+.7+.8+.9)/4

Once the assignments are worth different amounts, you don't really have four of anything anymore, so you don't divide by four to get the average. If one assignment (say, the one the student got an 80% on) was worth twice as much as any other, then it would be worth 40 points and all the others would be worth 20 points. The weighted average calculation would give you
(20*.6+20*.7+40+.8+20*.9)/(20+20+40+20)=20(.6+.7+2*.8+.9)/20*5, so you'd effectively be averaging five grading units, because the one on which the student scored an 80 counts as two grading units. To average the resulting grading units, a 60%, a 70%, *two* 80%s, and a 90%, you'd add up those five numbers and divide by five -- the "four"ness is all gone, because there really aren't four grading units being averaged, but rather five.

Bethbethbeth's approach below, looking at everything as a multiple of 5, effectively works because she's got a base grading unit of 5%. The assignment worth 5% counts as one grading unit, the one worth 10% counts as two grading units, the one worth 15% counts as three, and so on. She's adding up all the score with multiplicity (ie, multiplying by the number of grading units that assignment is worth). In the end, a score on the 20% assignment gets added in four times, and so on. She then divides by the resulting total number of effective grading units -- 20 -- to get the average of all 20 of the grading units, each worth 5%.

Doing it will percentages out of 100 is the same thing all over again, where a single "grading unit" is one point, and you can think of "an assignment worth 5%" as being effectively counting as 5 grading units. There are a total of 100 grading units, and dividing by 100 is basically invisible when you're talking about percentages.

I know this is really long-winded, but my impression is that you're trying to figure out how the percentage-based weighted average relates to the good old "add up all the numbers and divide by how many there are" average that you use if everything's worth the same amount. I hope my explanation helps; it's a lot easier to explain when I can judge how clear I'm being with feedback and facial expressions!

[tzikeh.livejournal.com]

my impression is that you're trying to figure out how the percentage-based weighted average relates to the good old "add up all the numbers and divide by how many there are" average that you use if everything's worth the same amount.

YES! That is what I am doing. The hard part is that I haven't taken a real math class in 20 years, so my commutative concepts are rusty, if not non-existent.

[lamardeuse.livejournal.com]

If you did it like that, you'd have to divide the result by 10, because then you're weighting the marks according to your base 10 scale.

[bethbethbeth.livejournal.com]

10%, 15%, 10%, 15%, 20%, 25%, and 5%.

Lowest common denominator is 5.

Score = X

10% = 2X
15% - 3X
10% = 2X
15% - 3X
20% = 4X
25% = 5X
5 % = 1X

The total points need to be divided by 20 (2 + 3 + 2 + 3 + 4 + 5 + 1)

So...if the scores were all 80s and 90's (in order)

10$ = 2X = 2x80 = 160
15% = 3X = 3x90 = 270
10% = 2X = 2x80 = 160
15% = 3X = 3x90 = 270
20% = 4X = 4x80 = 320
25% = 5X = 5x90 = 450
5 % = 1X = 1x80 = 80

Total - 1710/20 (and therefore the final grade is 85.5%)

[tzikeh.livejournal.com]

okay - so, what made you approach this from the lowest-common-denominator-of-percentages direction? And why does it need to be divided by 20?

[bethbethbeth.livejournal.com]

I approached it that way because...um...I always have.

But it needs to be divided by 20 because that's the sum of the multiples of 5 (see this: (2 + 3 + 2 + 3 + 4 + 5 + 1)

[bethbethbeth.livejournal.com]

Are you trying to put together a lesson plan? Because my forte in this area is doing, not explaining (which is why I teach Lit classes and not math classes *g*)

[silveraspen]

Multiply each assignment's earned/achieved grade by the percentage weighting assigned to it, then sum all the results to come up with the total for the overall grade.

It may seem a little odd because it's not the standard "add all grades together and divide by the total number of assignments" method, but it's still accurate.

Let's see if I can articulate it differently. Say you had a student who scored perfectly (100 out 100 points on every assignment); breaking it down, it'd look like this:

Assign 1: 100 x 10% = 100 x 0.10 = 10 points
Assign 2: 100 x 15% = 100 x 0.15 = 15 points
Assign 3: 100 x 10% = 100 x 0.10 = 10 points
Assign 4: 100 x 15% = 100 x 0.15 = 15 points
Assign 5: 100 x 20% = 100 x 0.20 = 20 points
Assign 6: 100 x 25% = 100 x 0.25 = 25 points
Assign 7: 100 x 5% = 100 x 0.05 = 5 points
TOTAL: -------------------------> 100 points

Or in other words, each percentage weight multiplied by each earned grade gives you the number of points from that graded assignment that makes up that percent of the total value.

[castalianspring.livejournal.com]

To put it in algebraic form, I'd use:

.1a + .15b + .1 c + .15d + .2e + .25f + .05g = X

...where a, b, c, etc are your grades and X is the final grade. You could make it a bit simpler by combining the grades that have equal weight, as in:

.1(a+c) + .15(b+d) + .2e + .25f + .05g = X

The "lesson" would probably be a combination of simple linear equations and percentages.

[finabair.livejournal.com]

So x is different for each assignment, as are the weights? No way to get around having to mostly multiply each one separately, then, except of course for the ones with the same weighting.

fg=final grade
g1, g2, g3 etc. are your individual x values, the grades for each test. And I'm going to convert the percents to decimals for ease of computation. * is what I'll use for multiplication. So thus:

fg = .1*g1 + .15*g2 + .1*g3 + .15*g4 + .2*g5 + .25*g6 + .05*g7

It can be consolidated like by grouping the multipliers (percents converted to decimals) like this:

fg = .1*(g1+g3) + .15*(g2 + g4) +.2*g5 + .25*g6 + .05*g7

I'd be inclined to put it in an Excel spreadsheet like that, if I had a lot to do.

[darthfox.livejournal.com]

The fact that each assignment is worth a different percent of the final grade means that each percentage point of each assignment is worth a slightly different amount. So the way to do it easily is to work out how big the points on each assignment are. There's a common-denominator thing happening here.

If the student has earned x points per assignment out of 100, I'd multiply x by the percentage of the final grade that assignment is worth. So if you designate the assignments A through G, then 10xA + 15xB + 10xC + 15xD + 20xE + 25xF + 5xG = the final number of points the student has earned. Divide that by 10000, which is the final number of points that were available. (This is just a way of working the whole thing with whole numbers as long as possible.)

That made a lot more sense in my head, but maybe it makes sense out here too. :-)

[heresluck]

Since you have actual answers, I can be a literalist and say: Screw algebra, I make Excel do it. *g*

[tzikeh.livejournal.com]

LOL!!! Well, I'd probably get a math teacher to do it for me, because first I'd have to know how to set it up in Excel. :D Yay colleagues!

[heresluck]

Actually, I find setting it up in Excel to be quite easy, because it's just a matter of telling the spreadsheet which number should be which percent of the grade. But this has always been my problem with math: it's much easier for me to puzzle through concrete tasks than to generate abstractions. Which is why geometry and calculus (which are fundamentally spatial) are MUCH easier for me to wrap my head around than algebra (or for that matter arithmetic). I'm not very FAST at geometry or calculus (especially after lo these many years away from them), but I understand them.

[natmerc.livejournal.com]

One way to figure it out, is to figure out what's the number value for each assignment and then add them up. It's exactly the same as the percentage method, but shown differently. It's sort of anti-fractions, because you're not trying to find a common denominator and it looks/is wrong mathematically, but sometimes people find it easier to visualize and figure out.

But there are seven assignments, worth 10%, 15%, 10%, 15%, 20%, 25%, and 5%.

Say you get 80% on each assignment. Muliply the 80 by the percentage value to get: 8/10, 12/15, 8/10, 12/15, 16/20, 20/25, and 4/5

You then just take all the top numbers and add them together:
(8+12+8+12+16+20+4) = 80

In reality, you'd have different numbers, e.g. 60% on the 25% assignment would give 60*.25 = 15/25, etc.

And some people would then add the bottom ones together, which add up to 100, and that would be BAD, fraction math, but people like to do it.

[natmerc.livejournal.com]

And that's pretty much what lamardeuse said.

Vbscript to help

[zephiey.livejournal.com]

This is the text file of a vbscript that will do the calculations for you.

script language="vbscript" (add < > at the beginning and end of this line. I had to take it out in order for you to see it.)
Option Explicit
Dim exam1, exam2, exam3, exam4, exam5, exam6, exam7, exam1wt, exam2wt, exam3wt, exam4wt, exam5wt,exam6wt, exam7wt, FinalAverage

exam1 = Inputbox("What is your exam 1 score?")
exam2 = Inputbox("What is your exam 2 score?")
exam3 = Inputbox("What is your exam 3 score?")
exam4 = Inputbox("What is your exam 4 score?")
exam5 = Inputbox("What is your exam 5 score?")
exam6 = Inputbox("What is your exam 6 score?")
exam7 = Inputbox("What is your exam 7 score?")


exam1wt = exam1 * .10
exam2wt = exam2 * .15
exam3wt = exam3 * .10
exam4wt = exam4 * .15
exam5wt = exam5 * .20
exam6wt = exam6 * .25
exam7wt = exam7 * .05

FinalAverage = exam1wt + exam2wt + exam3wt + exam4wt + exam5wt + exam6wt + exam7wt

Document.write "Your final average will be " & FinalAverage

script (add </ at the beginning and > at the end to close the script)


Copy and paste it to Note or Wordpad, save it as an HTM file on your harddrive. Start up Internet Explorer, Go to FILE > OPEN> Browse for the HTM. Click OK and Accept the script warning to run Active X. Enter your grades and the script will do the work for you.

As you can see in the script it applies the percentage weight of each grade, multiplies the grade and percentage then adds them all up to achieve the Final Avg.

You can add or subtract exams in the script just make sure that for every one added or subtracted a place is put in or taken out of the DIM and the equation is written for it. Also edit the Final Average to reflect how many exams are added or subtracted.

Yes, I know more info than you wanted but I've just spent a semester writing vbscript for everything. You get to benefit from it.